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Dynamical systems on networks with adaptive couplings appear naturally in real-world systems such as power grid networks, social networks, and neuronal networks. We investigate a paradigmatic syste...

We consider rings of coupled phase oscillators with anisotropic coupling. When the coupling is skew-symmetric, i.e., when the anisotropy is balanced in a specific way, the system shows robustly a c...

We perform bifurcation analysis of plane wave solutions in a one-dimensional complex cubic-quintic Ginzburg--Landau equation with delayed feedback. Our study reveals how multistability and snaking behavior of plane waves emerge as time delay is introduced. For intermediate values of the delay, bifurcation diagrams are obtained by a combination of analytical and numerical methods. For large delays, using an asymptotic approach we classify plane wave solutions into strongly unstable, weakly unstable, and stable. The results of analytical bifurcation analysis are in agreement with those obtained by direct numerical integration of the model equation.

The Lang–Kobayashi model is a system of delay differential equations (DDEs) describing the dynamics of a semiconductor laser under delayed optical feedback. In this paper, we study the stability of so-called external cavity modes (ECMs), which are harmonic oscillations corresponding to stationary lasing states. We focus on experimentally relevant situations, when the delay is large compared to the internal time scales of the laser. In this case, both the number of ECMs and the number of critical eigenvalues is large. Applying a newly developed asymptotic description for the spectrum of linearized DDEs with long delay, we are able to overcome this difficulty and to give a complete description of the stability properties of all ECMs. In particular, we distinguish between different types of weak and strong instabilities and calculate bifurcation diagrams that indicate the regions with different stability properties and the transitions between them.